Issue 51

K. Bahram et alii, Frattura ed Integrità Strutturale, 51 (2020) 467-476; DOI: 10.3221/IGF-ESIS.51.35 468 Structures subjected to cyclic loadings can undergo large variations in their behaviour ranging from the phase of plastic deformation to breaking through damage depending on the nature of the stresses. The constant nature of these stresses at constant amplitude implies the use of laws and models which are mastered by the current computation codes, currently, the PARIS law is the expression most used in fatigue research work with constant amplitude, it should be noted that this relation is applicable only in the field of slow crack propagation (domain II) and it does not take into account what happens in domain I and III [4-6] . Nevertheless, the constant amplitude propagation laws make it possible to describe the propagation of long cracks under constant amplitude loading. Indeed, it is implicitly assumed that a given cycle causes a loss assimilated to an advance regardless of the history of previous loading. However, actually, the structures are only very rarely subjected to loads of constant amplitude. On the other hand, the load spectrum measurements indicate a variation of the stress amplitude over time. In addition, experience shows that the damage induced by a given cycle may depend on the previous history of loading [7-9]. It has now been found that the application of an overload cycle during a fatigue crack propagation leads to a slowing of the crack propagation speed, in some cases to a stop of this crack [10-12], this phenomenon of delay due to the application of an overload, several authors [13-15], proposed different models to explain the causes due to the slowing of the crack propagation speed. These different approaches can be grouped into three categories of models, based on: i) the interaction effects of the plastic areas at the tip of the crack. These models are inspired by that of Wheeler [14]. ii) the phenomenon of closing the crack due to the residual stresses induced by the plastification near the crack tip. These models are based on that of Elber [13]. iii) iii) micromechanisms that act at the tip of the crack. These models are inspired by [16-18], in the case of an elastoplastic material. The strongly deformed area near the crack. is governed by an oligocyclic fatigue mechanism. In the following, we will use the AFGROW calculation code and the model describing the delay of Willembourg to study the influence of the application of simple overload on different parameters such as, crack spread, crack rate and overload rate. W ILLENBORG MODEL he Willenborg model is a slightly different approach since it proposes to determine a slowing factor [15], but an actual value of the ratio of load at crack point eff R . The formulation of the generalized Willenborg model implemented in the AFGROW code is shown below: min max eff eff eff K R K  (1) with min eff min R K K K   (2) max eff max R K K K   (3) The value of the residual stress intensity factor R K is defined for a crack length i a , necessary to create a plasticized zone of size eq R , tangent the plasticized zone created by the overload R pic . min max min max max 0  0 ,  0  0 ,  0 ,  0                                            eff eff eff eff eff eff max eff eff eff if K and K K K if K and K K K if K K                   (4) The residual stress intensity factor is given by the following equation: T